Symmetry for Quasicrystals

1. Symmetry for Quasicrystals References: http://www.jcrystal.com/steffenweber/qc.html F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14. http://en.wikipedia.org/wiki/Icosahedral_symmetry http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/advanced-chemistryprize2011.pdf

2. Definition of Quasicrystals (QCs) Materials With perfect long-range order, but with no 3D translational periodicity. non-crystallographic rotational symmetry Sharp diffraction spots Old definition of Crystals Definition till 1991: A crystal is a solid where the atoms form a periodic arrangement.

3. New Definition for Crystal International Union of Crystallography, “Report of the Executive Committee for 1991”, ActaCryst., A48, (1992), 922. “ … By crystal, we mean any solid having an essentially discrete diffraction diagram, and by aperiodic crystal we mean any crystal in which three dimensional lattice periodicity can be considered to be absent” Diffraction Pattern  crystals !

4. Periodicity Order Crystals   Quasicrystals X  Amorphous X X Crystals Quasicrystals Translation, t inflation,  Rotation 1, 2, 3, 4, 6 Rotation 1, 2, 3, 4, 5, 6, 8, 10, 12 τ : scaling ratio

5. Types of QCs Quasiperiodicin 2D (polygonal or dihedral QCs, one periodic direction  the quasiperodiclayers) Octagonal QCs: local 8-fold symmetry [P & I] Decagonal QCs: local 10-fold symmetry [P] Dodecagonal QCs: local 12-fold symmetry [P] Quasiperiodicin 3D (no periodic direction) Icosahedral QCs: (axes:12x5-fold, 20x3-fold, 30x2-fold) [P, I & F] new type (reported in Nature, Nov.2000) “Icosahedral" QCs with broken symmetry (stable binary Cd5.7Yb)

6. Octagonal QCs Chris J. Pickard and R. J. Needs, Nature Materials 9,624–627

7. Decagonal QCs http://nanopatentsandinnovations.blogspot.tw/2011/10/quasicrystals-discovery-wins-novel.html

8. Dodecagonal QCs http://www.pnas.org/content/108/5/1810/F6.expansion.html

9. http://en.wikipedia.org/wiki/File:Icosahedron.gif Schematic drawings of the unit cell of fcc Zr2Ni structure (a) and examples of icosahedral clusters around Zr and Ni atoms in the unit cell (b). J. Saida et al., Intermetallics, V. 10, Issues 11–12, November 2002, Pages 1089–1098 Icosahedral QCs

10. Simulations of some diffraction patterns F. Samavat et. al., Open Journal of Physical Chemistry, 2012, 2, 7-14. A simulation from an icosahedral quasicrystal

11. http://www.lassp.cornell.edu/lifshitz/quasicrystals.html  4  3  2 

12. Example of 1D QCs

13. Cut and Project Fibonacci sequence (1D QCs) HaraldBohr, ActaMathematicae,45, 580 (1925) Make a cut in a 2D space and project the mathematical points onto a 1D space, a line, and get a 1D quasicrystal Ignore anything outside of the two lines   tan irrational number (why?) Choose  E.g. : Make cuts in a 6D space and project in 3D space  3D QCs Fibonacci number

14. Aperiodic Periodic Aperiodic crystal Periodic crystal ~ approximant (called)

15. Fibonacci number (series, sequence) Fibonacci Rabbits: Fibonacci’s Problem: If a pair of new born rabbits are put in a pen, how many pairs of rabbits will be in the pen? Assumptions: 1. Can produce once every month 2. Always produce one male and one female offspring 3. Can reproduce once they are one month old 4. The rabbits never die

16. continue Birth 1st month Grow up 2nd month 3rd month 4th month 5th month 6th month 1 2 4 3 6 5 7 Month 8 # of pairs 1 2 3 5 8 1 13 ? 21

17. Fibonacci number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ….. The sequence Fn of Fibonacci numbers is defined by the recurrence relation Golden ratio

18. A B B A B A B B BA B BA BA B BA B BA BAB BA BAB BAB BA BAB BA BAB 1-D QC

21. Ho-Mg-Zn Quasicrystal from http://cmp.physics.iastate.edu/canfield/photos.html